Gamma inversa

Inverse Gamma Distribution
John D. Cook October 3, 2008

Abstract These notes write up some basic facts regarding the inverse gamma distribution, also called the inverted gamma distribution. In asense this distribution is unnecessary: it has the same distribution as the reciprocal of a gamma distribution. However, a catalog of results for the inverse gamma distribution prevents having torepeatedly apply the transformation theorem in applications. Here we derive the distribution of the inverse gamma, calculate its moments, and show that it is a conjugate prior for an exponentiallikelihood function.

1

Parameterizations

There are at least a couple common parameterizations of the gamma distribution. For our purposes, a gamma(α, β) distribution has density f (x) = 1 xα−1exp(−x/β) Γ(α)β α

for x > 0. With this parameterization, a gamma(α, β) distribution has mean αβ and variance αβ 2 . Define the inverse gamma (IG) distribution to have the density f (x) = for x > 0. β α −α−1 xexp(−β/x) Γ(α)

1

2

Relation to the gamma distribution

With the above parameterizations, if X has a gamma(α, β) distribution then Y = 1/X has an IG(α, 1/β) distribution. To see this,apply the transformation theorem. d −1 y dy

fY (y) = fX (1/y) = =

1 y −α+1 exp(−1/βy) y −2 Γ(α)β α (1/β)α −α−1 y exp(−(1/β)/y) Γ(α)

3

Moments

Next we calculate the moments of X ∼ IG(α, β).If α > n, E(X n ) = = = = = In particular, for α > 1 E(X) = and for α > 2 E(X 2 ) = and so for α > 2 V ar(X) = E(X 2 ) − E(X)2 = β2 . (α − 1)2 (α − 2) β α−1
∞ βα xn x−α−1 exp(−β/x) dx Γ(α) 0 ∞ βαxn−α−1 exp(−β/x) dx Γ(α) 0 β α Γ(α − n) Γ(α) β α−n β n Γ(α − n) (α − 1) · · · (α − n)Γ(α − n) βn . (α − 1) · · · (α − n)

β2 (α − 1)(α − 2)

2

4

Conjugate prior for exponential likelihoodFinally, suppose than an observation X | µ ∼ exponential(µ) and a priori µ ∼ IG(α, β). By Bayes’ theorem, the posterior distribution on µ given an observation X = x is proportional to 1 1 1...